Foundations of Science

, Volume 22, Issue 4, pp 763–783 | Cite as

Toward a History of Mathematics Focused on Procedures

  • Piotr Błaszczyk
  • Vladimir Kanovei
  • Karin U. Katz
  • Mikhail G. Katz
  • Semen S. Kutateladze
  • David Sherry


Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework.


Tangent Line Ultimate Foundation Infinitesimal Quantity Expert Defence Shaky Ground 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We are grateful to Paul Garrett for subtle remarks that helped improve an earlier version of the text. M. Katz was partially funded by the Israel Science Foundation grant number 1517/12.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Institute of MathematicsPedagogical University of CracowKrakówPoland
  2. 2.IPPIMoscowRussia
  3. 3.MIITMoscowRussia
  4. 4.Department of MathematicsBar Ilan UniversityRamat GanIsrael
  5. 5.Sobolev Institute of MathematicsNovosibirsk State UniversityNovosibirskRussia
  6. 6.Department of PhilosophyNorthern Arizona UniversityFlagstaffUS

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